Analytical optimization of spread and change of exponential decay of generalized Wannier functions in one dimension

Abstract

Generalized Wannier functions of a couple of bands in a one-dimensional crystal are investigated. A lower bound for the global minimum of the total spread is obtained. Assumption of such a value being the minimum leads to a first-order differential equation for the transformation matrix. Simple analytical solutions leading to real generalized Wannier functions are presented for consecutive bands in a crystal with inversion symmetry. Results are displayed for a particle in a diatomic Kronig-Penney potential. For the lowest couple of bands, calculated single-band Wannier functions resemble orbitals of a diatomic molecule, whereas generalized Wannier functions seem like orthogonalized atomic orbitals. The latter functions are neither symmetric nor antisymmetric and display increased exponential decay. For the next two pairs of bands, Wannier functions retain their centers and symmetries, and exponential decay does not increase. Results are also shown to be in agreement with solutions of the eigenvalue problem of the band-projected position operator.